On invertible algebras

When doing arithmetic in most familiar situations, such as with integers, we can always add, subtract, and multiply. However, depending on the setting, we cannot always divide. For example, if we're only working with integers, the only numbers we can always divide by are 1 and -1. Note, however, that any number can be written as a sum of such invertible elements, for example 3 = 1 + 1 + 1. A similar situation arises in linear algebra: one can add, subtract, and multiply (square) matrices, but not necessarily divide. However, Wolfson and Zelinksy independently proved in the 1950s that any matrix can be written as a sum of precisely two invertible matrices. The study of mathematical structures that are generated by their invertible elements has since become an active area of research in ring theory.

Invertible algebras are a recent development in this area. An algebra is a mathematical structure having an addition and multiplication that generalizes the integers. An element u of an algebra is invertible if there exists some v such that u ˑ v = v ˑ u = 1. This work focuses on invertible algebras, algebras where every element can be written as a sum of invertible elements, and the related notion of I2 algebras: algebras where each element may be written uniquely as a sum of invertible elements and as a sum of their inverses. Any I2 algebra is tautologically invertible, but the converse is not clear: given an invertible algebra, is it necessarily I2? We answer this question for various types of algebras, and additionally show that a broad class of invertible algebras are “almost I2” in a precise mathematical sense.